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WEE LIANG GAN AND LIPING LI

Abstract. The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps.

We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion ofd-step central stability, and prove that if the ideal of relations of a category is generated in degrees at mostd, then every module presented in finite degrees isd-step centrally stable.

1. Introduction

LetRbe a commutative ring. For any small categoryC, aC-moduleoverRis, by definition, a (covariant) functor from C to the category ofR-modules. Amorphism ofC-modules overR is, by definition, a natural transformation of functors. The category ofC-modules overR is an abelian category. One can define, in a natural way, the notion of afinitely generated C-module overR.

Let F I be the category of finite sets and injective maps, and denote by Z+ the set of non-negative integers. For anyF I-moduleV overR andn∈Z+, we write Vn forV([n]), where [n] :={1, . . . , n}. Since the automorphism group of [n] in the category F I is the symmetric group Sn, an F I-module V over R gives rise to a sequence{Vn} whereVn is a representation ofSn. It was shown by Church, Ellenberg, Farb, and Nagpal (see [3] and [4]) that many interesting sequences of representations of symmetric groups arise in this way from finitely generatedF I-modules. A principle result they proved is that any finitely generated F I-module over a commutative noetherian ring is noetherian. They deduced as a consequence that if V is a finitely generated F I-module over a commutative noetherian ring, then the sequence {Vn} admits an inductive description, in the sense that for all sufficiently large integerN, one has

(1.1) Vn∼= colim

S⊂[n]

|S|6N

V(S) for eachn∈Z+,

where the colimit is taken over the poset of all subsetsS of [n] such that |S|6N.

To provide the context for our present article, let us briefly describe the way in which (1.1) is proved in [4]. Suppose thatV is a finitely generatedF I-module over a noetherian ring. For any finite setT, there is a canonical homomorphism ofR-modules

(1.2) colim

S⊂T

|S|6N

V(S)−→V(T).

SinceV is finitely generated, there exists an integerN0such thatV is generated byti6N0Vi. It is easy to see that the homomorphism (1.2) is surjective ifN>N0. To prove the injectivity of (1.2) whenN is sufficiently large, Church, Ellenberg, Farb, and Nagpal constructed a (Koszul) complexSe−∗V ofF I-modules which has the property that:

(H1(Se−∗V))(T) = Ker

colim

S⊂T

|T|−26|S|6|T|−1

V(S)→V(T) .

They verified that one has:

(1.3) colim

S⊂T

|T|−26|S|6|T|−1

V(S) = colim

S⊂T

|S|<|T|

V(S).

2010Mathematics Subject Classification. 16P40; 16G99.

Key words and phrases. Central stability, representation stability, FI-modules, finitely presented.

The second author is supported by the National Natural Science Foundation of China 11541002, the Construct Program of the Key Discipline in Hunan Province, and the Start-Up Funds of Hunan Normal University 830122-0037. Both authors would like to thank the anonymous referee for carefully checking the manuscript and providing quite a few valuable comments.

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From the noetherian property ofV, they proved that there exists an integerN00such that (H1(Se−∗V))(T) = 0 if|T|>N00. As a consequence,

(1.4) colim

S⊂T

|S|<|T|

V(S) =V(T) if|T|>max{N0, N00}.

SetN = max{N0, N00}. The homomorphism in (1.2) is an isomorphism if|T|6N, for T is terminal in the poset{S|S ⊂T}. It now follows by an induction on|T|that (1.2) is an isomorphism too if|T|> N, for:

colim

S⊂T

|S|6N

V(S) = colim

U⊂T

|U|<|T|

colim

S⊂U

|S|6N

V(S) = colim

U⊂T

|U|<|T|

V(U) =V(T),

where the first isomorphism is routine, the second isomorphism is by the induction hypothesis, and the third isomorphism is by (1.4).

It was subsequently proved by Church and Ellenberg that for anyF I-moduleV over an arbitrary com- mutative ring, if V is presented in finite degrees (in a suitable sense), then there are integers N0 and N00 such thatV is generated byti6N0Vi, and (H1(Se−∗V))(T) = 0 if |T|>N00. Consequently, using the same arguments as above, they showed that the isomorphism (1.1) holds for allN sufficiently large. This extends the result of their joint work with Farb and Nagpal to F I-modules which are not necessarily noetherian (since every finitely generatedF I-module over a commutative noetherian ring is presented in finite degrees).

The left-hand side of (1.3) is isomorphic to the central stabilization construction of Putman [11, §1].

In their joint work, Putman and Sam generalized the isomorphism in (1.1) to modules over complemented categories with a generator of whichF I is an example. A complemented category with a generatorX is the data of a symmetric monoidal category and an objectX satisfying a list of axioms. Their proof is similar to the one forF I described above.

A goal of our present paper is to give a very simple and transparent proof of a generalization of the above results. In particular, our proof does not require consideration of the complex Se−∗V or similar complexes.

Moreover, the setting for our generalization is much simpler than the one of complemented categories with a generator used by Putman and Sam [12], and include many more examples; our generalization is, in fact, motivated by the fact that several combinatorial categories studied by Sam and Snowden in [14] do not fall within the framework of [12]. Another goal of our paper is to explain the role of the quadratic property of F I in the isomorphism (1.3).

Outline of the paper. This paper is organized as follows.

In Section 2, we define our generalization of the notion of central stability and introduce the notion of d-step central stability. We show that in the special case of complemented categories with a generator studied by Putman and Sam, our notion of central stability is indeed equivalent to their notion of central stability.

We also show that for the categoryF I, it is equivalent to the inductive description (1.1).

In Section 3, we give a reminder of a key lemma from Morita theory. We then prove our first main result, that a module is presented in finite degrees if and only if it is centrally stable. We deduce that if every finitely generated module is neotherian, then every finitely generated module is centrally stable. We prove that the converse of the preceding statement holds under a local finiteness assumption on the category when the base ring is a commutative noetherian ring. To illustrate the wide applicability of our results, we recall examples of combinatorial categories introduced by Sam and Snowden in [14] which fall within our framework but are not complemented categories with a generator.

In Section 4, we prove our second main result, that if the ideal of relations of a category is generated in degrees at most d, then every module presented in finite degrees is d-step centrally stable; for example, if the category is quadratic, then every module presented in finite degrees is 2-step centrally stable.

To apply our second main result, one needs to have a practical way to check that the ideal of relations of a category is generated in degrees at mostd. In Section 5, we give sufficiency conditions which allow one to do this.

2. Generalities

2.1. Notations and definitions. We prefer to formulate our main results in the more familiar language of modules over algebras. Throughout this paper, we denote by R a commutative ring andZ+ the set of non-negative integers.

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LetAbe anR-linear category, i.e. a category enriched over the category ofR-modules. We assume that Ob(A) =Z+ and HomA(m, n) = 0 ifm > n. We set

(2.1) A:= M

m,n∈Ob(A)

HomA(m, n).

There is a natural structure of a (non-unital)R-algebra onA. (In the terminology of [1],Ais aZ-algebra.) We callAthecategory algebra ofA. For eachn∈Ob(A), we denote byen the identity endomorphism ofn.

For anym, n∈Ob(A) withm6n, we set

em,n:=em+em+1+· · ·+en ∈A.

One has: Aem,n=Aem⊕Aem+1⊕ · · · ⊕Aen.

A graded A-module is an A-module V such that V = L

n∈Ob(A)enV. Observe that if V is a graded A-module, then anyA-submodule ofV is also a gradedA-module.

A graded A-module V is finitely generated if it is finitely generated as an A-module. Equivalently, V is finitely generated if for some N ∈ Ob(A), there is an exact sequence L

i∈IAe0,N →V → 0 where the indexing setIis finite. A gradedA-moduleV isnoetherianif everyA-submodule ofV is finitely generated.

A gradedA-moduleV isfinitely presentedif for someN ∈Ob(A), there is an exact sequence of the form M

j∈J

Ae0,N −→M

i∈I

Ae0,N −→V −→0 where both the indexing setsIandJ are finite.

A gradedA-moduleV ispresented in finite degrees if for someN ∈Ob(A), there is an exact sequence

(2.2) M

j∈J

Ae0,N −→M

i∈I

Ae0,N −→V −→0

where the indexing setsI andJ may be finite or infinite. The smallestN for which such an exact sequence exists is called thepresentation degree ofV, and we denote it by prd(V).

Remark 2.1. It is easy to see that if V is a graded A-module presented in finite degrees, then for every N >prd(V), there exists an exact sequence of the form (2.2), whereIandJ may depend onN.

The main definitions of this paper are as follows.

Definition 2.2. A graded A-moduleV is calledcentrally stable if for allN sufficiently large, one has

(2.3) Ae⊗eAeeV ∼=V wheree=e0,N.

Definition 2.3. Letdbe an integer>1. A gradedA-moduleV is calledd-step centrally stable if for allN sufficiently large, one has

Ae⊗eAeeV ∼= M

n>N−(d−1)

enV where e=eN−(d−1),N.

2.2. Inductive descriptions. We now explain the relation of our definition of central stability given above to the notion of central stability given by Putman and Sam in [12] for complemented categories with a generator, and the relation to the inductive description (1.1) in the special case ofF I-modules as given by Church, Farb, Ellenberg, and Nagpal [2, 4].

LetC be a small category such that Ob(C) =Z+, and HomC(m, n) =∅ ifm > n. We denote byAC the R-linear category with Ob(AC) =Z+ and HomAC(m, n) the freeR-module with basis HomC(m, n) for each m, n∈Z+. LetAC be the category algebra ofAC; see (2.1).

Recall that aC-module overRis, by definition, a (covariant) functor fromCto the category ofR-modules.

For anyC-moduleV overRandn∈Z+, we writeVnforV(n). IfV is aC-module overR, thenL

n∈Ob(C)Vn

is a gradedAC-module. This defines an equivalence from the category ofC-modules overR to the category of graded AC-modules. We say that aC-moduleV over Rhas a certain property (such as centrally stable) if the gradedAC-moduleL

n∈Ob(C)Vn has the property.

For anyM, N ∈Ob(C) with M 6N, we write CM,N for the full subcategory of C on the set of objects {M, M+1, . . . , N}. Then the category ofCM,N-modules overRis equivalent to the category ofeACe-modules

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where e=eM,N. LetιM,N :CM,N → C be the inclusion functor. We define therestriction functor ResM,N alongιM,N by

ResM,N : (category of C-modules overR)−→(category ofCM,N-modules overR), V 7−→V ◦ιM,N.

Theleft Kan extension functor LanM,NalongιM,Nis a left adjoint functor to ResM,N; for everyCM,N-module V overRand n∈Ob(C), one has:

(2.4) (LanM,NV)n= colim

α:s→n M6s6N

Vs,

where Vs = V(s) and the colimit is taken over the (comma) category whose objects are the morphisms α:s→nin Csuch that M 6s6N; see [7, Theorem 2.3.3] or [10, Section X.3, (10)].

The following proposition gives a reformulation for the notions of central stability and d-step central stability forC-modules overR.

Proposition 2.4. Let V be a C-module over R. For everyM, N ∈Ob(C)with M 6N, one has:

(2.5) M

n∈Ob(C)

(LanM,N(ResM,NV))n ∼=Ae⊗eAee M

n∈Ob(C)

Vn

whereA=AC and e=eM,N. In particular,V is centrally stable if and only if for all sufficiently largeN, one has

Vn∼= colim

α:s→n s6N

Vs for each n∈Z+.

Moreover,V isd-step centrally stable if and only if for all sufficiently largeN, one has Vn∼= colim

α:s→n N−(d−1)6s6N

Vs for eachn>N−(d−1).

Proof. Lete=eM,N. For anyC-moduleV overR, one has M

M6n6N

(ResM,NV)n=e M

n∈Ob(C)

Vn

.

Since the functor:

(category ofeAe-modules)−→(category of gradedA-modules), W 7−→Ae⊗eAeW,

is left adjoint to the functor:

(category of gradedA-modules)−→(category ofeAe-modules), V 7−→eV,

it follows that for anyCM,N-moduleW overR, one has M

M6n6N

(LanM,NW)n∼=Ae⊗eAe

M

M6n6N

Wn

.

We have proven the isomorphism (2.5). The remaining statements now follow from the formula (2.4).

A complemented category with a generator, in the sense of Putman and Sam [12], is the data of a symmetric monoidal category A and an object X of A satisfying a list of axioms. We do not recall those axioms here since we will not need them; however, a consequence of the axioms is that the full subcategory C of Aon the set of objectsXn for n∈Z+ is a skeleton ofA, and HomC(Xm, Xn) = ∅ ifm > n. We may identify the set of objects ofCwithZ+in the obvious way. Following Putman and Sam [12, Theorem E], an A-moduleV overRiscentrally stable if for all sufficiently largeN, the functorV is the left Kan extension of the restriction ofV to the full subcategory of Aspanned by the objects isomorphic toXn for somen6N. Corollary 2.5. Let A be a complemented category with a generator X. Suppose that C is the skeleton ofA spanned by the objectsXnfor all n∈Z+. LetV be anA-module overR, and regardV also as aC-module by restriction along the inclusion functor C →A. Then V is centrally stable in the sense of Putman and Sam [12, Theorem E]if and only if V is centrally stable as aC-module overR.

Proof. This is immediate from (2.5).

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The full subcategory ofF I spanned by the objects [n] for alln∈Z+ is a skeleton ofF I. In the following corollary, we identify the set of objects of this skeleton withZ+ in the obvious way.

Corollary 2.6. Suppose that C is the skeleton of F I spanned by the objects[n] for alln∈Z+. Let V be anF I-module overR, and regard V also as a C-module by restriction along the inclusion functorC → F I.

ThenV is centrally stable as aC-module over Rif and only if for all sufficiently large N, the isomorphism (1.1)holds.

Proof. By Proposition 2.4, it suffices to show that colim

α:s→n s6N

Vs∼= colim

S⊂[n]

|S|6N

V(S).

But this is immediate from the observation that the natural functor:

{α|α∈HomF I([s],[n]) ands6N} −→ {S|S ⊂[n] and |S|6N}, α7−→Im(α),

is final; see [10, Theorem IX.3.1]. (Some authors refer to final functors as cofinal functors. See, for example,

[7, Definition 2.5.1].)

Let us also mention that for a principal ideal domainR, Dwyer defined the notion of acentral coefficient system ρ in [6] to mean a sequence ρn of GLn(R)-modules and maps Fn : ρn → ρn+1 such that Fn is a GLn(R)-map (when ρn+1 is considered as a GLn(R)-module by restriction) and the image of Fn+1Fn is invariant under the action of the permutation matrix sn+2 ∈ GLn+2(R) which interchanges the last two standard basis vectors of Rn+2. Suppose that C is the skeleton of F I spanned by the objects [n] for all n ∈ Z+. A central coefficient system ρ defines a C-module V over R with Vn = ρn and such that the standard inclusion [n],→[n+ 1] induces the mapFn. IfV is finitely generated as aC-module overR, then by [4, Theorem C] it is a centrally stableC-module overR.

3. Central stability

3.1. Key lemma. The following lemma is a standard result in Morita theory; see, for instance, [5, Theorem 6.4.1]. We recall its proof here since it plays a key role in the proof of our main results.

Lemma 3.1. LetAbe any (possibly non-unital) ring, andean idempotent element ofA. IfV is anA-module such that, for some indexing setsI andJ, there is an exact sequence

(3.1) M

j∈J

Ae−→M

i∈I

Ae−→V −→0, thenAe⊗eAeeV ∼=V.

Proof. Applying the right-exact functor Ae⊗eAe e(−) to (3.1), we obtain the first row in the following commuting diagram:

(3.2) L

j∈JAe⊗eAeeAe //

L

i∈IAe⊗eAeeAe //

Ae⊗eAeeV //

0

L

j∈JAe //L

i∈IAe //V //0

Both the rows in (3.2) are exact. Since the two leftmost vertical maps are isomorphisms, the rightmost

vertical map is also an isomorphism.

3.2. Central stability and presentation in finite degrees. LetA be an R-linear category such that Ob(A) =Z+ and HomA(m, n) = 0 ifm > n. Denote byAthe category algebra of A; see (2.1).

Theorem 3.2. Let V be a graded A-module. Then V is presented in finite degrees if and only if it is centrally stable. Moreover, if V is presented in finite degrees, then the isomorphism (2.3)holds if and only if N>prd(V).

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Proof. Suppose thatV is presented in finite degrees, andN >prd(V). Then there is an exact sequence (2.2) withe=e0,N; see Remark 2.1. Hence, by Lemma 3.1, one hasAe⊗eAeeV ∼=V.

Conversely, suppose that N is an integer such that Ae⊗eAe eV ∼= V, where e = e0,N. There is an exact sequence L

j∈JeAe →L

i∈IeAe→ eV → 0 for some indexing setsI and J. Applying the functor Ae⊗eAe(−), we obtain an exact sequence of the form (2.2) withe=e0,N. Therefore,V is presented in finite

degrees andN >prd(V).

Corollary 3.3. If every finitely generated gradedA-module is noetherian, then every finitely generated graded A-module is centrally stable.

Proof. IfV is a finitely generated graded A-module, then V is finitely presented, hence centrally stable by

Theorem 3.2.

The above proofs are very simple in comparison to the proofs for the special cases given in [2, Theorem B], [4, Theorem C], and [12, Theorem E]. Let us emphasize, however, that the crux is to recognize that the notion of central stability in [2, 4, 12] can be reformulated in Morita theory;this was not a priori obvious.

3.3. Central stability and noetherian property. We prove a partial converse to Corollary 3.3.

Let A be an R-linear category such that Ob(A) = Z+ and HomA(m, n) = 0 if m > n. Denote by A the category algebra of A; see (2.1). We say that A is locally finite if HomA(m, n) is a finitely generated R-module for all m, n∈Z+.

Corollary 3.4. Suppose that R is a commutative noetherian ring and A is locally finite. If every finitely generated gradedA-module is centrally stable, then every finitely generated graded A-module is noetherian.

Proof. LetV be a finitely generated graded A-module and U a (graded) A-submodule of V. We want to prove thatU is finitely generated as a gradedA-module. SinceV /U is a finitely generated gradedA-module, it is centrally stable, hence presented in finite degrees by Theorem 3.2. Thus, for someN ∈Ob(A), there is a commuting diagram

L

j∈JAe0,N //

g

L

i∈IAe0,N //

f

V /U //0

0 //U //V //V /U //0.

It suffices to prove that both Im(g) and Coker(g) are finitely generated gradedA-modules.

By the Snake Lemma, there is an isomorphism Coker(g)∼= Coker(f), so Coker(g) is a finitely generated gradedA-module.

SinceAis locally finite andV is finitely generated as a gradedA-module, it follows thate0,NV is a finitely generatedR-module, hence e0,NV is a noetherianR-module. But Im(g) is generated as a gradedA-module bye0,NIm(g), ande0,NIm(g) is anR-submodule of the noetherianR-modulee0,NV, hence Im(g) is finitely

generated as a gradedA-module.

3.4. Examples. Let us mention some examples of combinatorial categories introduced by Sam and Snowden [14] which fall within our framework. The central stability of modules presented in finite degrees for these categories follow from Theorem 3.2 (and were not previously known).

In the following examples, the set of objects of the category C is Z+. For any n ∈ Z+, we set [n] :=

{1, . . . , n}.

(i) C = F Ia where a is an integer > 1. For any m, n ∈ Z+, a morphism m → n in F Ia is a pair (f, c) where f : [m] → [n] is an injective map and c : [n]\Im(f) → [a] is any map. The composition of (f1, c1) :m→nand (f2, c2) :n→`is defined to be (f2◦f1, c) :m→` wherec(i) =c1(j) ifi=f2(j) for somej ∈[n]\Im(f1), andc(i) =c2(i) ifi∈[`]\Im(f2).

(ii)C =OIa wherea is an integer>1. For any m, n∈Z+, a morphismm→nin OIa is a pair (f, c) where f : [m] →[n] is a strictly increasing map and c : [n]\Im(f)→ [a] is any map. The composition of morphisms is defined in the same way as above forF Ia.

(iii) C =F Sop, the opposite of the categoryF S. For anym, n ∈ Z+, a morphism n → m in F S is a surjective map [n]→[m]. The composition of morphisms inF S is defined to be the composition of maps.

It was proved by Sam and Snowden [14] that for the categoriesCin (i)-(iii) above, every finitely generated gradedC-module over a commuatative noetherian ring is noetherian (and hence they are centrally stable by Corollary 3.3).

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4. d-step central stability

4.1. Ideal of relations. SupposeV is a finitely generatedF I-module over a commutative noetherian ring.

The isomorphism (1.1) says that, providedN is sufficiently large,V can be recovered from its restriction to the full subcategory ofF I on the set of objects{S∈Ob(F I)| |S|6N}; using the isomorphism (1.3), we see that in fact we can recoverV(S) for |S|> N using only the restriction ofV to the full subcategory of F I on the set of objects {S ∈ Ob(F I) |N −1 6|S|6N}. The purpose of the present section and the next is to show that this is a consequence of the quadratic property ofF I. More generally, we shall prove that if the ideal of relations of a category algebraA is generated in degrees6d (in the sense of Definition 4.1 below), then any gradedA-module which is presented in finite degrees isd-step centrally stable.

LetAbe an R-linear category such that Ob(A) =Z+ and HomA(m, n) = 0 if m > n. Denote byAthe category algebra ofA; see (2.1). For anym∈Z+, let

A(m, m) := Ende A(m);

for anyn>m+ 1, let

A(m, n) := Home A(n−1, n)⊗EndA(n−1)· · · ⊗EndA(m+1)HomA(m, m+ 1).

We define aR-linear categoryAewith Ob(A) =e Z+by Hom

Ae(m, n) =A(m, n) ife m6n, and Hom

Ae(m, n) = 0 ifm > n. There is a naturalR-linear functor fromAeto Awhich is the identity map on the set of objects.

For anyn>m, letI(m, n) be the kernel of the mape A(m, n)e →HomA(m, n). One hasI(m, n) = 0 whenevere nismorm+ 1.

Definition 4.1. Letdbe an integer>1. We say thatthe ideal of relations of Ais generated in degrees6d if, whenevern>m+ 2, the mapA(m, n)e →HomA(m, n) is surjective, and whenevern>m+d, one has:

(4.1) I(m, n) =e

n−d

X

r=m

A(re +d, n)⊗EndA(r+d)I(r, re +d)⊗EndA(r)A(m, r).e

Remark 4.2. Ifd= 1, so that the ideal of relations ofAis generated in degrees 61, thenI(m, n) = 0 fore everym andn, and soAeandAare the same. In most interesting examples, one hasd>2.

Remark 4.3. Ifd= 2, so that the ideal of relation ofA is generated in degrees62, thenA is a quadratic algebra whose degreekcomponent is M

m∈Z+

HomA(m, m+k) for eachk∈Z+. Many combinatorial categories such asF Ia,OIa andF Sopare quadratic; see Section 5 below.

Example 4.4. Fix a finite totally ordered set Ω. Recall (see [8]) that the plactic monoidM on Ω is the monoid generated by Ω with defining relations

xzy=zxy ifx6y < z, yxz=yzx ifx < y6z.

An elementw∈M is said to be of length`(w) =nifwis a product ofnelements of Ω; it is clear that`(w) is well-defined. Now defineC to be the category with Ob(C) =Z+ and

HomC(m, n) ={w∈M |`(w) =n−m}.

The composition of morphisms in C is given by the product in M. Then the category algebra AC is not quadratic but has ideal of relations generated in degrees63.

It is plain that the preceding example can be generalized to any monoid with a presentation whose defining relations do not change the length of words.

4.2. d-step central stability. The following theorem is the second main result of this paper.

Theorem 4.5. Let dbe an integer >1. Suppose the ideal of relations of A is generated in degrees6d. If a graded A-moduleV is presented in finite degrees, then V isd-step centrally stable.

To prove Theorem 4.5, we need the following lemma.

Lemma 4.6. Suppose the ideal of relations of A is generated in degrees 6d. IfV is a gradedA-module, andn > N >m+d, then

enAf⊗f Aff V ∼=enAe⊗eAeeV wheree=em,N, f =em+1,N.

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Proof. Observe that

enAe=enAem⊕enAf, eV =emV ⊕f V.

We have a natural map

Φ :enAf⊗f Af f V →enAe⊗eAeeV.

We shall construct a map Ψ inverse to Φ. First, define the maps

I(m, n)e ⊗RemV

µ

HomA(m+ 1, n)⊗EndA(m+1)HomA(m, m+ 1)⊗RemV eθ //

ν

enAf⊗f Aff V

HomA(m, n)⊗RemV

whereµandνare the obvious maps defined using composition of morphisms, andθeis defined byeθ(α⊗β⊗x) = α⊗βx. The mapν is surjective and its kernel is the image ofµ. Using (4.1) and HomA(m+ 1, m+d)⊂f Af, we see thateθµ= 0. Therefore,θefactors uniquely throughν to give a map

θ: HomA(m, n)⊗RemV →enAf⊗f Aff V.

Now define

Ψ :e enAe⊗ReV →Af⊗f Aff V by

Ψ(αe ⊗x) =

θ(α⊗x) ifα∈enAemandx∈emV, α⊗x ifα∈enAf andx∈f V,

0 otherwise.

It is plain thatΨ descends to a mape

Ψ :enAe⊗eAeeV →Af⊗f Aff V,

and Ψ is an inverse to Φ.

We can now prove Theorem 4.5.

Proof of Theorem 4.5. By Theorem 3.2, for allN sufficiently large, one has Ae0,Ne0,NAe0,N e0,NV ∼=V; in particular, for eachn∈Z+, one hasenAe0,Ne0,NAe0,N e0,NV ∼=enV.

Ifn > N >d−1, then by Lemma 4.6,

enAeN−(d−1),NeN−(d−1),NAeN−(d−1),N eN−(d−1),NV ∼=enAeN−d,NeN−d,NAeN−d,N eN−d,NV ...

∼=enAe0,Ne0,NAe0,N e0,NV IfN−(d−1)6n6N, the map

enAeN−(d−1),NeN−(d−1),NAeN−(d−1),N eN−(d−1),NV →enV, α⊗x7→αx

has an inverse defined byx7→en⊗x.

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5. Sufficiency conditions

5.1. Sufficiency conditions. To apply Theorem 4.5, we need to be able to check if the ideal of relations ofAis generated in degrees6d. In this section, we provide sufficiency conditions which allow one to check this.

LetC be a small category such that Ob(C) =Z+, and HomC(m, n) =∅ifm > n. Recall thatAC is the R-linear category with Ob(AC) =Z+ and HomAC(m, n) the freeR-module with basis HomC(m, n) for each m, n∈Z+. LetAC be the category algebra ofAC; see (2.1).

Definition 5.1. We say that theideal of relations of C is generated in degrees6dif the ideal of relations ofAC is generated in degrees6d.

Proposition 5.2. Supposed> 2. The ideal of relations of C is generated in degrees 6d if the following two conditions are satisfied:

(i) The composition map HomC(l, n)×HomC(m, l)→HomC(m, n) is surjective wheneverm < l < n.

(ii) For everyα1, α2∈HomC(m+ 1, n)andβ1, β2∈HomC(m, m+ 1) satisfying α1β12β2 and n > m+d,

there existsγ∈HomC(m+d, n)andδ1, δ2∈HomC(m+ 1, m+d)such that the following diagram commutes:

(5.1) m+ 1

δ1 $$

α1

++m

β1 <<

β2 ""

m+d γ //n

m+ 1

δ2

::

α2

33

Proof. Condition (i) implies thatAeC(m, n)→HomAC(m, n) is surjective ifn>m+ 2. We shall prove, by induction onn−m, that (4.1) holds. The casen−m=dis trivial. Now supposen−m > d. Let

AeC(m, n)−→π1 HomAC(m+ 1, n)⊗End

AC(m+1)HomAC(m, m+ 1)−→π2 HomAC(m, n)

be the obvious maps defined by composition of morphisms. It is easy to see (and can be proved in the same way as [9, Lemma 6.1]) thatI(m, n) is spanned overe R by elements of the form

(5.2) ξn−1⊗ · · · ⊗ξm−ξ0n−1⊗ · · · ⊗ξ0m

such thatξi, ξi0 ∈HomC(i, i+ 1) for i=m, . . . , n−1 andξn−1· · ·ξmn−10 · · ·ξm0 . By condition (ii), there existsγ∈HomC(m+d, n) andδ1, δ2∈HomC(m+ 1, m+d) such that

ξn−1· · ·ξm+1 =γδ1, ξn−10 · · ·ξ0m+1=γδ2, δ1ξm2ξ0m. We can choose

eγ=eγn−1⊗ · · · ⊗eγm+d∈AeC(m+d, n),

whereeγi∈HomC(i, i+ 1) fori=m+d, . . . , n−1, such thateγn−1· · ·eγm+d=γ. We can also choose δe=δem+d−1⊗ · · · ⊗eδm+1∈AeC(m+ 1, m+d),

whereeδi∈HomC(i, i+ 1) fori=m+ 1, . . . , m+d−1, such thatδem+d−1· · ·δem+11. Similarly, choose δe0 =eδm+d−10 ⊗ · · · ⊗δe0m+1 ∈AeC(m+ 1, m+d),

whereeδi0 ∈HomC(i, i+ 1) fori=m+ 1, . . . , m+d−1, such thateδ0m+d−1· · ·δe0m+12. The element in (5.2) can be written as:

n−1⊗ · · · ⊗ξm+1−eγ⊗eδ)⊗ξm−(ξn−10 ⊗ · · · ⊗ξm+10 −eγ⊗eδ0)⊗ξm0 +eγ⊗(eδ⊗ξm−eδ0⊗ξm0 ).

Since

ξn−1⊗ · · · ⊗ξm+1−γe⊗δe∈I(me + 1, n), ξ0n−1⊗ · · · ⊗ξ0m+1−γe⊗eδ0 ∈I(me + 1, n),

eδ⊗ξm−eδ0⊗ξm0 ∈I(m, me +d),

the result follows by induction.

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Corollary 5.3. Suppose that Csatisfies the two conditions in Proposition 5.2 for somed>2. If aC-module V is presented in finite degrees, then V isd-step centrally stable.

Proof. By Proposition 5.2, the ideal of relations of C is generated in degrees 6d. Hence, we may apply

Theorem 4.5.

Remark 5.4. Suppose thatCsatisfies condition (i) in Proposition 5.2, and its ideal of relations is generated in degrees6d. In this case, condition (ii) might not hold, so it is not a necessary condition. For example, suppose thatd= 2, and one has

HomC(0,1) ={β1, β2, β3}, HomC(1,2) ={β10, β20, β30, β04}, HomC(2,3) ={β100, β200}, with the defining relations

β10β130β3, β20β204β3, β001β30200β40, as depicted in the following diagram:

1 β

0

1 //2

β100

&&

0

β1

88

β2

&&

β3 //1

β30

88

β40

&&

3

1 β20

//2

β200

88

Letα1100β10 andα2002β02. Then one has

α1β1100β10β1100β30β3200β40β3200β20β22β2.

On the other hand, it is easy to see that there do not existγ,δ1 andδ2such that the diagram in (5.1) (with d= 2,m= 0,n= 3) commutes.

5.2. Examples. Using Proposition 5.2, one can easily check, for example, that (a skeleton of) F I is qua- dratic. Let C be the full subcategory of F I on the set of objects {[n] | n ∈ Z+}. It is easy to see that C satisfies condition (i) in Proposition 5.2. We need to verify condition (ii). Thus, suppose that we have injective maps:

α1, α2: [m+ 1]−→[n] and β1, β2: [m]−→[m+ 1], such thatα1◦β12◦β2 andn>m+ 2. Since

|Im(α1)∪Im(α2)|=|Im(α1)|+|Im(α2)| − |Im(α1)∩Im(α2)|6(m+ 1) + (m+ 1)−m=m+ 2, we can choose S ⊂ [n] such that Im(α1)∪Im(α2) ⊂S and |S| = m+ 2. Letγ : [m+ 2] → [n] be any injective map whose image isS. Then there exists a unique map δ1 (respectively δ2) such that γ◦δ11 (respectivelyγ◦δ22). Clearly,δ1andδ2 are injective. We have:

γ◦δ1◦β11◦β12◦β2=γ◦δ2◦β2.

Since γ is injective, it follows that δ1◦β12◦β2. Therefore, it follows from Proposition 5.2 thatC (or more precisely, the algebraAC) is quadratic.

Similarly, one can apply Proposition 5.2 to show that the categories F Ia, OIa, F Sop, and VI(F) are quadratic, where for any fieldF, the set of objects ofVI(F) isZ+and the morphismsm→nare the injective linear mapsFm→Fn.

Remark 5.5. A twisted commutative algebraEis an associative unital graded algebraE=L

n>0Enwhere eachEnis equipped with the structure of anSn-module such that the multiplication mapEn⊗Em→En+m

is Sn×Sm-equivariant, and yx =τ(xy) for everyx ∈En, y ∈ Em (where τ ∈Sn+m switches the first n and lastmelements of{1, . . . , n+m}); see [13,§8.1.2]. Any twisted commutative algebraEgives rise to an R-linear categoryAwith Ob(A) =Z+and

HomA(m, n) =RSnRSn−mEn−m.

In particular, when E is the twisted commutative algebra with En the trivialSn-module for every n(and with the obvious multiplication map), the R-linear category A we obtained is precisely the category AC

associated to a skeletonCofF I. One might ask if the category algebra of every categoryAobtained from

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a twisted commutative algebra is quadratic. This is not true. For example, let dbe any integer >2, and letE be the twisted commutative algebra withEn the trivialSn-module ifn < d, otherwise letEn= 0; the multiplication map of E is defined in the obvious way. Then E gives rise to anR-linear categoryA with the property that the ideal of relations of its category algebra is generated in degrees6d, but the category algebra is not quadratic.

References

[1] A. Bondal, A. Polishchuk, Homological properties of associative algebras: the method of helices, Russian Acad. Sci. Izv.

Math. 42 (1994), no. 2, 219-260.

[2] T. Church and J. Ellenberg, Homology of FI-modules, to appear in Geo. Topo., arXiv:1506.01022v1.

[3] T. Church, J. Ellenberg, B. Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833-1910, arXiv:1204.4533v4.

[4] T. Church, J. Ellenberg, B. Farb, R. Nagpal, FI-modules over Noetherian rings, Geom. Top. 18-5 (2014), 2951-2984, arXiv:1210.1854v2.

[5] P.M. Cohn, Algebra, Vol. 3, Second edition, John Wiley & Sons, Ltd., Chichester, 1991.

[6] W.G. Dwyer, Twisted homological stability for general linear groups, Ann. of Math. (2) 111 (1980), no. 2, 239-251.

[7] M. Kashiwara, P. Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, 332, Springer- Verlag, Berlin, 2006.

[8] A. Lascoux, B. Leclere, J. Y. Thibon, The plactic monoid, in Algebraic Combinatorics on Words, ed. M. Lothaire, Ency- clopedia of Math. and Appl., Vol. 92, Cambridge University Press, 2002, pp. 164-196.

[9] L. Li, A generalized Koszul theory and its application, Trans. Amer. Math. Soc. 366 (2014), no. 2, 931-977, arXiv:1109.5760v3.

[10] S. Mac Lane, Categories for the working mathematician. Second edition. Graduate Texts in Math., 5. Springer-Verlag, New York, 1998.

[11] A. Putman, Stability in the homology of congruence subgroups, Invent. Math. 202 (2015), no. 3, 987-1027, arXiv:1201.4876v5.

[12] A. Putman, S. Sam, Representation stability and finite linear groups, arXiv:1408.3694v2.

[13] S. Sam, A. Snowden, Introduction to twisted commutative algebras, arXiv:1209.5122.

[14] S. Sam, A. Snowden, Gr¨obner methods for representations of combinatorial categories, J. Amer. Math. Soc. 30 (2017), 159-203, arXiv:1409.1670v3.

Department of Mathematics, University of California, Riverside, CA 92521, USA E-mail address:wlgan@math.ucr.edu

Key Laboratory of Performance Computing and Stochastic Information Processing (Ministry of Education), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China

E-mail address:lipingli@hunnu.edu.cn

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